تعادل بازار معمالات آتی با عدم تجانس و بازار لحظه ای در برداشت محصول

This paper studies equilibrium in the futures market for a commodity in a single good economy, which is populated by heterogeneous producers and speculators. The commodity is traded only in the spot market at harvest whereas futures contracts written on the commodity are traded continuously. The model illustrates the role of heterogeneity and non-tradeness in a futures market equilibrium. The results show that the futures price is driven by aggregate wealth, rather than the spot price as in other models and that the futures price process is a simple one which depends on the relative risk process.

This paper studies hedging and (partial) equilibrium in the futures market for a commodity which is populated by heterogeneous producers and speculators. The commodity is traded in a spot market only at harvest time. Each producer is endowed with a non-traded private technology and trades in futures contracts in order to reduce her quantity and price risks. Speculators invest their initial wealth in bonds and take positions in futures contracts written on the commodity. The model is motivated by the observation that a spot market is open only at harvest time for some commodities.1 In such a setting, the producer faces both a price risk and a quantity risk. However, most of the optimal hedging literature deals only with price risk. Hirshleifer, 1990 and Hirshleifer, 1991 discusses the effects of both types of risk on the hedging decision but does not model them simultaneously. A simple way to represent the quantity and price risks of each producer is to model them as a private cash flow risk, where the cash position of a hedger is the present value of her terminal cash flows. The difficulty in solving the equilibrium under these conditions is that both the futures price and the cash position of a hedger are endogenous. In more traditional models, the cash position, which is the fixed quantity of the commodity held by the producer, is independent of the futures price.2 The optimal demand for a futures contract depends on tastes, on the resolution of uncertainty, and on the formulation of the hedging problem. Early contributions in the optimal hedging literature usually assumed that hedgers were monoperiodic expected utility maximizers (see for instance Stein, 1961; Johnson, 1960; Anderson and Danthine, 1980; Losq, 1982). A common result from most of these papers is that the optimal hedge ratio consists of a pure hedge component, and a mean variance component. More recent models are derived in a continuous-time framework, whereby the hedger maximizes the expected utility of intertemporal consumption subject to a wealth-budget or cash-budget dynamic constraint. The cash-budget formulation has been used by Ho (1984) in a model in which a farmer, subject to both output and price risk during the production period, hedges a non-traded position. The optimal demand for futures contracts depends on the exogenous cash position (the output), and includes both a mean-variance efficiency component and a Merton–Breeden dynamic hedging component. The wealth-budget formulation, initiated by Stulz (1984), is used by Adler and Detemple (1988) to solve a problem similar to that of Ho. The resulting demand for futures contracts depends on the output, but also includes a mean-variance efficiency term, a dynamic hedging component, and a minimum-variance component. For myopic investors, both formulations yield tractable optimal demands, similar to those obtained in the single period models. Therefore, investors are assumed to have logarithmic utility functions in the model developed herein. The demand for futures contracts depends on the uncertainty structure of the spot and the futures market. A producer that hedges against price risk to protect a non-traded position has a demand dependent on the output. Without trade constraints, and in the absence of frictions, the demand for futures contracts is independent of the output (see Briys et al., 1990). In the model developed herein, a producer is concerned with both quantity and price risks, and thus hedges against the risk of individual cash flows. Although equilibrium futures price and portfolio policies are determined simultaneously, the determination herein proceeds in two stages for mathematical convenience. First, the demand of each agent is derived assuming that the parameters of the futures price are known. Then, individual demands are aggregated to obtain the equilibrium futures price at any point in time. The optimal demand for futures contracts by each investor is derived by assuming temporarily that the parameters of the futures price are known, and by applying the equivalent martingale method developed by Karatzas et al. (1987). Diversity in cash flow risk leads to different hedging strategies. The demand for futures contracts by a producer depends on her wealth, the present value of her terminal production, and the relative risk process of the futures contract. This structure of demand is motivated by a desire to diversify, and to insure the actual value of the terminal production. Producers are long or short in the futures contract if there is heterogeneity in cash flow risk. A similar result is obtained by Hirshleifer (1991) in a discrete-time model where heterogeneity in the resolution of uncertainty leads to different hedging strategies. In the model developed herein, the differential in cash flow risk can also be interpreted as a differential in the resolution of uncertainty. Once the optimal demand for futures contracts is derived, the aggregation follows easily. A continuous-time equilibrium exists in the futures market when producers account for both quantity and price risks by hedging against their individual and terminal cash flow risks. Stochastic cash positions and equilibrium futures price are endogenous, unlike previous equilibrium models which ignore quantity risk. The futures price depends only on the volatility of the aggregate wealth. The volatility of the futures price is stochastic and increases, on average, as the time to maturity decreases. This maturity effect in the model is consistent with Samuelson (1965). The stochastic volatility obtained in the model is also consistent with the recent empirical evidence on the behaviour of daily futures prices.3 The positive mean of the instantaneous growth rate of futures prices indicates normal backwardation in the model. The relative risk process of futures prices is a weighted sum of the present values of all the terminal productions of hedgers. Aggregation is made easier by assuming a logarithmic utility function. The relative risk process of futures prices in the model of heterogeneous agents developed herein has a weighted-average structure. Detemple and Murty (1994), who appear to be the first to deal with heterogeneity in an intertemporal production economy, obtain a similar result for the equilibrium interest rate.4 Our results show that the futures price is driven by aggregate wealth, rather than the spot price as in other models and that the futures price process is a simple one which depends on the relative risk process itself. Contrary to the contango hypothesis (a futures market dominated by risk averse commodity producers) or the normal backwardation (a futures market dominated by commodity users), our model shows that producers could be long or short on the futures. Another interesting result which is consistent with that of Anderson and Danthine (1983) and the Samuelson hypothesis, and the empirical observations of Milonas (1986), is that volatility in futures prices increases as time to maturity shortens. Finally, our results are obtained while allowing for the simultaneous resolution of both price and quantity risks. The remainder of the paper is organized as follows. In Section 2, the economy is described. In Section 3, optimal demands for futures contracts by each type of investor are derived. In Section 4, the equilibrium futures price is derived; and in Section 5, some concluding remarks are offered.

Hedging and equilibrium are studied herein for a setting in which: (1) heterogeneous producers hedge to reduce their respective terminal cash flow risks, which emanate from non-traded production technologies, by trading in futures contracts written on the commodity they grow, and (2) speculators invest their initial wealth in bonds and trade in futures contracts. The distinctive feature of the model is that cash flow (and not price) risk drives trades in futures contracts. Consequently, the futures price and the cash position are endogenous. Furthermore, equilibrium prices are derived when both quantity and price risks can be hedged simultaneously. This theoretical framework yields some interesting insights. First, the net position in futures contracts depends on the relative volatility of the producers’ cash flows. Producers with cash flow uncertainty, which are resolved less rapidly than the aggregate, will tend to go short in order to smooth their terminal consumption, or alternatively, will hold futures contracts. This result differs from that obtained in traditional models in which producers sell futures contracts to hedge their non-traded positions against spot price fluctuations. This result is consistent with the casual observation that the distinction between hedgers and speculators is ambiguous, and thus producers can be both on the long and the short sides. Second, changes in the equilibrium futures price depend on wealth and the relative risk process. The futures price dynamics resulting from this model are consistent with empirical evidence (Milonas, 1986) for commodity futures price, where trading is absent. Specifically, the model predicts an inverse relation between volatility of futures price increments and maturity (the Samuelson hypothesis): Volatility increases as time to maturity shortens. An interesting avenue for future research would be to study a similar model in an incomplete market setting. This might account better for the ARCH behaviour reported in studies of futures prices.